Step 1Heat the oven to 375 degrees. Coat 2 baking sheets with nonstick spray and set aside.
Step 2Beat together the butter and sugar in a large mixing bowl until fluffy. Add the eggs, one at a time, beating after each. Add the milk and vanilla and lemon extracts and mix until smooth.
Step 3In a medium bowl, combine the cake and all-purpose flours, the baking powder and the salt and stir until mixed. Add the dry ingredients to the wet in batches, mixing well to combine. For each cookie, place 1/4 cup of batter on a baking sheet. Use a knife or offset spatula to shape each cookie into a circle; do not flatten. Space the cookies 2 inches apart.
Step 4Bake the cookies until the edges begin to brown, 15 to 17 minutes. Remove to a wire rack to cool completely.
Step 1Place the powdered sugar in a large bowl. Gradually add enough of the boiling water, stirring constantly, until the mixture is thick and spreadable.
Step 2Remove half of the frosting to a double boiler set over, but not touching, simmering water and add the chocolate. Warm the mixture, stirring, until the chocolate is melted and the frosting is smooth. Remove from the heat.
Step 3Turn the cookies to frost the bottoms. Using a knife or offset spatula, coat half of each cookie with chocolate frosting and the other half with white frosting. Repeat until all cookies are done.